In general relativity, spacetime (an object which unifies the classical Newtonian notions of space and time) is a four-dimensional pseudo-Riemannian manifold of signature (1,3) or (3,1).
"Space" is now a term that we may assign to any so-called spacelike hypersurface within this manifold - although there may be pathological cases, one may think of such a hypersurface as a "time slice" in the sense that all the events in it are, at least from the viewpoint of some observers, simultaneous but at a spatial distance.
The expansion of space through the FLRW scale factor simply means that if you take two points in such a hypersurface that are at a certain distance and evolve the "space" forward in time, you'll find that the two corresponding points on the "future" spatial hypersurface are at a larger distance than that. This is expansion of space - if you designate some spatial volume and evolve it forward in time, its volume will grow. The notion of time evolution I am thinking of here is rooted in the initial value formulation of general relativity, where we start with a spatial hypersurface as initial data and evolve it forward in time to get a spacetime that is foliated by the time-evolved versions of this hypersurface.
As an analogy, you may think about points on the surface of an expanding ballon. We are looking at a three-dimensional spacetime, that is given by the union of the ballon surface at each instant of time. Space is two-dimensional here, it is given by the ballon surface at a fixed time. Mark two points on the surface of a ballon. As you blow up the ballon, the distance between them (measured as the shortest path on the surface of the ballon connecting them) increases. The only flaw in this analogy is that it is not relativistic - it's not clear what changing observers would correspond to, but it carries well the idea of time-evolution of a (hyper)surface and of increasing distance.
It is important to note that a particle that is at one place on the past surface is not necessarily carried to the corresponding point in the future surface - bound systems do not expand when space expands, see this question and its answers. Nowhere has matter entered in this discussion, both the notion of space and the notion of space expansion are purely geometric and make - at least theoretical - sense in an empty universe, although you will of course have a hard time observing space expansion in an empty universe. In the balloon analogy, two things you affix to the surface on the balloon and which are connected by a rubber band will stay at the same distance as the balloon expands, while the purely geometrical marked points increase in their distance.